In this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

In this paper we determine the smallest equivalence relation on a multialgebra for which the factor multialgebra is a universal algebra satisfying a given identity. We also establish an important property for the factor multialgebra (of a multialgebra) modulo this relation.

In this paper we investigate when negative definite functions on commutative hypergroups satisfy the Schoenberg criterion.

In this paper we consider Fourier multipliers for $L^p$ $(p>1)$ on Chébli-Trimèche hypergroups and establish a version of Hörmander's multiplier theorem. As applications we give some results concerning the Riesz potentials and oscillating multipliers.

1991 Mathematics Subject Classification: 43A62, 43A15, 43A32.

In this paper we consider Fourier multipliers on local Hardy spaces ${{\mathbf{h}}^{\mathbf{p}}}(0<p\le 1)$ for Chébli-Trimèche hypergroups. The molecular characterization is investigated which allows us to prove a version of Hörmander’s multiplier theorem.